https://file.aiquickdraw.com/mathbot/file/1725761293357ntwyur9b.jpg

To find the missing side length \( n \), we can use the concept of similar triangles. Since the two triangles in the image are similar (i.e., their corresponding angles are the same), the ratio of the corresponding sides will be equal.

In this case, we have:

- Small triangle: \(\overline{AB} = 2\), \(\overline{AC} = 4\), \(\overline{BC} = 5\)
- Larger triangle: \(\overline{RQ} = n\), \(\overline{QS} = 10\), \(\overline{RS} = 12.5\)

Since the triangles are similar, the ratio of corresponding sides is equal:

\[
\frac{\overline{AB}}{\overline{RQ}} = \frac{\overline{AC}}{\overline{QS}} = \frac{\overline{BC}}{\overline{RS}}
\]

Using the known sides, we can set up a proportion:

\[
\frac{2}{n} = \frac{4}{10}
\]

Now, we can solve for \( n \):

\[
2 \times 10 = 4 \times n
\]

\[
20 = 4n
\]

\[
n = \frac{20}{4} = 5
\]

Thus, the missing side length \( n \) is 5.

Would you like further clarification or more details on any step?

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Here are five related questions to expand your understanding:

1. How do we know that the two triangles are similar?
2. How can the Pythagorean theorem be used to verify the side lengths of a right triangle?
3. What are other properties of similar triangles that can be useful in solving problems?
4. How can proportions be used in different geometric figures aside from triangles?
5. How would the solution change if the given values for sides were different?

**Tip:** Always verify if the triangles are similar by comparing the corresponding angles before applying the proportion rule.

Find the missing side length n in two similar triangles, where the smaller triangle has sides 2, 4, and 5, and the larger triangle has sides 10, 12.5, and n.

Learn how to find the missing side length n in a pair of similar triangles using proportions. The smaller triangle has sides 2, 4, and 5, and the larger triangle has sides 10, 12.5, and n. Suitable for students in Grades 7-9.

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